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Section7.4Arrow's Theorem, Conclusions and Exercises

So Wheres the Fair Method?

Here is a summary of whether each of our voting methods satisfies each of our fairness criteria:

Majority Criterion Monotonicity Criterion Condorcet Criterion IIA Criterion
Plurality Method Satisfies Satisfies Violates Violates
Borda Count Method Violates Satisfies Violates Violates
Instant Runoff Voting (IRV) Method Satisfies Violates Violates Violates
Pairwise Comparison Method Satisfies Satisfies Satisfies Violates
Table7.41Fairness criteria satisfied by each voting method

So far none of the voting methods we have studied satisfy our four fairness criteria. At this point, you are probably asking why we keep looking at method after method just to point out that they are not fully fair. We must be holding out on the perfect method, right? Unfortunately, no.

To see a very simple example of how difficult voting can be, consider the election below:

Number of votes 5 5 5
1st choice A C B
2nd choice B A C
3rd choice C B A
Table7.42A paradoxical election

Notice that in this election:

10 people prefer A over B
10 people prefer B over C
10 people prefer C over A
Table7.43Results of paradoxical election

No matter whom we choose as the winner, 2/3 of voters would prefer someone else! This scenario is dubbed Condorcets Voting Paradox, and demonstrates how voting preferences are not transitive (just because A is preferred over B, and B over C, does not mean A is preferred over C). It also demonstrates the impossibility of identifying a winner this election -- because there is no fair resolution.

Not only is it true that none of our voting methods satisfy all of the fairness criteria we care about, even worse, it turns out that there is no perfect voting method.

SubsectionArrow's Impossibility Theorem

In 1949, a mathematical economist named Kenneth Arrow was able to prove that there is no voting method that will satisfy all the fairness criteria we have discussed.

Arrow's Impossibility Theorem

Any voting method that uses voters' ranked choices to decide an election with a single winner will violate at least one of the four criteria we've discussed: majority, monotonicity, Condorcet, or IIA.

In other words, Arrows theorem means that no matter how hard we try to find one, we cannot find a voting method using ranked choices to find a winner that satisfies every fairness criteria we have studied here. Consequently, in any election a losing candidate can claim the outcome is unfair in some way and be correct. This puts democracies in a difficult position: how can a winner of an election be determined in a manner that all constituents believe is consistently fair?

The answer is, it cant. Instead, we change the question and focus on finding a voting method that satisfies the most or the most important fairness criteria. Interestingly, the fact that there is no single fairest method might explain why there are many different voting methods used throughout the world.

This map is based on the one found here. It has been modified by combining similar systems into common categories and making the colors easier to distinguish.

Figure7.44Electoral systems around the world

The map in Figure7.44 shows the systems used to elect the lower house of each country's legislature. You can see a larger version if you right-click and select view image. Note that the Borda count is included in the other systems category (green), but is used in only one of those countries: the Pacific island nation of Nauru. It is also used, however, to elect two seats in the parliament of Slovenia that are reserved for the Hungarian and Italian ethnic minorities.

There are so many different voting methods, including ones we haven't discussed in this chapter, and all have their pros and cons. Fairness criteria can't tell us which method is the best, but they can at least provide some useful information to consider. It is then up to a given community to consider the many options thoughtfully and chose one that will help it make the best collective decisions that it can.

SubsectionChapter 7 Exercises

  1. Use the results of Election 1 as shown in the table to answer the questions below.

    Number of votes 26 25 15 34
    1st A B B C
    2nd C A C A
    3rd B C A B
    Table7.45Preference Table for Election 1

    1. How many total voters were there?
    2. True or False: a total of 26 voters preferred Candidate A over Candidate B.
    3. Who is the plurality winner? Does that candidate have a majority?
  2. Calculate the winner of Election 1 using the Borda count.

  3. The table below shows presidential election results for Florida in 2000, between George W. Bush, Al Gore, and Green Party candidate Ralph Nader. The vote totals, given as percentages, are the real results. 6Simplified to exclude candidates with less than 1% of the vote. The voters' second and third choices are educated guesses based on what we know of voters' opinions during that election.

    Percent of votes 49.2 49.1 1.7
    1st Bush Gore Nader
    2nd Gore Nader Gore
    3rd Nader Bush Bush
    Table7.46Preference Table for 2000 U.S. Presidential Election in Florida

    1. In this example, does a Condorcet candidate exist? If so, who is it?
    2. What (if anything) does this election tell us about the plurality method and the Condorcet criterion?

  4. Calculate the winner of Election 2 (shown below) using IRV.

    Number of votes 17 15 5 10 2 1
    1st A B C C D D
    2nd D C A D C B
    3rd B A D B B A
    4th C D B A A C
    Table7.47Preference Table for Election 2

  5. The table below shows the ballots for Election 3. It is the same as Election 2, except that some voters changed their first place votes from A to C.

    Number of votes 5 12 15 5 10 2 1
    1st C A B C C D D
    2nd A D C A D C B
    3rd D B A D B B A
    4th B C D B A A C
    Table7.48Preference Table for Election 3

    1. Calculate the winner of Election 3 using IRV.
    2. Compare Election 2 and Election 3. Do these elections show IRV violating one of the fairness criteria? If so, which one, and why?